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2010 NDIA GROUND VEHICLE SYSTEMS ENGINEERING AND TECHNOLOGY SYMPOSIUM MODELING & SIMULATION, TESTING AND VALIDATION (MSTV) MINI-SYMPOSIUM AUGUST 17-19 DEARBORN, MICHIGAN

SURVIVABILITY ENHANCED RUN-FLAT VARIABLE FOOTPRINT

James Capouellez Dr. Abraham Pannikottu US Army RDE-COM American Engineering Group TARDEC, AMSRD-TAR-D, MS 233 934 Grant St, Suite #101, Warren, MI 48397-5000 Akron, OH 44311, USA

Dr. Jon Gerhardt American Engineering Group 934 Grant St, Suite #101, Akron, OH 44311, USA

ABSTRACT The military has a unique requirement to operate in different terrains throughout the world. The ability to travel in as much varying terrain as possible provides the military greater tactical options. This requirement/need is for the to provide a variable footprint to allow for different . Much of the current run-flat technology utilized by the military severely limits mobility and adds significant weight to the unsprung mass. This technology gap has allowed for the development of new run- technology. New tire technology (fig 1) has been developed that substantially increases survivability, eliminates the need for heavy run-flat inserts, significantly reduces air pressure requirements and provides full (or near full) speed capability in degraded/damaged mode (punctured tire). This run-flat technology is built directly into the tire, yet maintains the normal variable footprint of a normal pneumatic tire. This makes the tire/wheel assembly much lighter and far more survivable than normal military run-flat technology. Safety, logistics, economics, and fuel economy are additional benefits this tire technology provides over current military tires with run-flat inserts.

INTRODUCTION The current state of the art tire technology utilizes a run-flat insert sometimes weighing over 100lbs. Fig 2 shows a cut-away showing the run-flat for a military tire/wheel. The stability of the vehicle running on a run-flat is significantly reduced and the operator is limited to a maximum speed of 30 mph, and sometimes less depending on stability. The distance that a vehicle can travel with a run-flat tire is also limited to typically 30 miles. This is also very similar to the NATO standard of 50 km [1]. The purpose of the run-flat is to allow the vehicle to have limp home capability and that’s it. Logistically the run-flat has further drawbacks. Unless you buy the wheel, run-flat, and tire as a package, you have to get the run-flat into the tire. This requires

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a special tool to compress profile encountered for different missions and the run-flat and get it into theaters, plus run-flat capability. New military the tire [2]. One such tool wheeled vehicles systems such as the Stryker or M- used on a HMMWV tire is ATV have on-board CTIS systems to enable these shown in Figure 3 [2]. The vehicles to quickly transition from highway mode to larger the run-flat the more off-road, or vice versa, within the safety of the difficult this becomes. This vehicle [3]. Most of the commercial market needs process further adds to the only one tire pressure setting for highway usage. maintenance Automobiles benefit the most from proper tire Figure 2. Military Run-flat burden on the inflation pressure, resulting in premium fuel Tire with Insert [1] military that has economy, extended tire life and proper handling. limited resources. This has lead to the development of stiff sidewall tires that provide similar handling and perform with Although run flat technology has been around for a or without air. However, the tire reliability and while, very little improvement has been made over durability are significantly worse without air the past 25-30 years. The current requirement of the pressure. Consumers of these tires have actually run-flat tire for the HMMWV is the same as it has complained that when the tire lost air pressure they ever been. had no degradation of performance so they kept running the tires until they were damaged beyond repair. This problem is the reason why some cars have tire pressure monitors to notify the driver of a flat tire. This is a far different experience of riding on a tire with a run-flat insert.

The stiff sidewall tire requires a very low aspect ratio so the tire can be supported properly by the sidewalls. “The sidewalls can't be very tall, so most Figure 3. HMMWV Run-flat are low-profile designs. Because of this, they are Insert Special Tool and typically used on sports, though they're also available Installation [2] for regular passenger cars and even minivans.”[4] The drawback with this tire is that the stiff sidewalls To make a run-flat insert better than what is tend to put more energy into the vehicles suspension currently produced is extremely difficult. The system and ultimately the vehicle. For new car commercial application for run-flat inserts is limited designs, this can be designed up front. However, the because of cost and performance. The systems military has vehicles that are in inventory for typically costs a lot more because of the additional extensive periods of time and cost to swap out hardware of the insert and the wheel needs to be suspension systems along with the tires can be a cost specially designed for the run-flat insert. Still, the that is hard to justify. Since the runflat is built into a premise of the run-flat insert is running on a small stiff sidewall, how much can the footprint can solid tire. The insert can only be so large or it will changed to provide different ground pressure for act as a bump stop for the tire and the width of the different terrain may be an issue that prevents this run-flat is going to be significantly less than the technology from being used for the military. width of the tire. This limits the surface area that the load can be distributed over. Also, the flat tire must either flop along or degrade due to the heat buildup of running directly on the tire and run-flat insert. This could lead to stability issues, smoke and possible flammability issues. There is a way to sum up the difficulty in making the run-flat performance significantly better, physics.

Aside from physics, another reason the development will be slow for the run-flat insert is that the commercial market has different requirements. Figure 4. Comparison of Conventional Tire to a Most military vehicles need to have the capacity to Stiff Sidewall Tire [4] change the tire pressure for the different terrain Unclassified/Distribution A Approved for Public Release Unclassified/Distribution A Approved for Public Release

To improve the runflat capability of the United weight of the springs could be reduced making the States military legacy systems, technology needs to concept more desirable. be developed that provides a variable footprint, can operate at zero air pressure with the carcass severely damaged/punctured, provide the same dynamic deflection as the normal pneumatic tire, and provide similar tread life. Could a tire be developed that would provide the same ground pressure requirements as a normal pneumatic tire such that it could meet the military demands of operating in sandy soft soil with zero or minimum air pressure, add air pressure for proper highway speeds and handling, and yet be able to provide enhanced run- flat capability. Could all this capability be developed in the tire to reduce the logistical burden associated with the run-flat inserts? From modeling, simulation, FEA, and laboratory testing the answer appears to be yes. Figure 5. Load and Boundary Conditions Applied on the Model for a First Order Foot Print Analysis Tire Development Further prototype development required integrating How does the pneumatic tire work? The load for the run-flat support into the tire. This would require the axle is resisted by the road. The body plies of the the development of rings or cylinders imbedded in tire between the axle and the road have reduced the tire. The shape of the ring could be optimized tension (resulting in the tire bulge). The cords above utilizing CAD and FEA analysis. the axle are in higher tension and as a result pull up The balance for the ring is going to have to be the axle. The means of putting the cords in tension achieving proper bonding between the ring and the is through air pressure. What other means could be tire rubber yet having the stiffness to support a pre- utilized to put the cords in tension. determined load with a desired change in the vertical diameter to produce the desired ground pressure.  Stiff Sidewalls Formulas and analysis for circular rings is much  Spring Molded Tire more straight forward.  Spring Inserted Tire Assuming no deformation, the change in height is  Hoop Ring Molded Tire (Metal) as follows:  Hoop Ring Molded Tire (Fiber Glass Rods) (1) WR 3  Carbon Fiber Molded Tire (Layer)  Dh   0 . 1488 [5]  Hoop Ring Molded Tire (Carbon Fiber) EI

To simplify development and test the concept, The ring can fail by either excessive compressive spiral springs were developed and inserted into the forces or by buckling. The critical stress in the ring tire. This was done on a small automotive tire to for buckling can be estimated by replacing the ring as quickly and economically evaluate the concept and a cylinder and doing the stress equation for a hollow help correlate stress analysis conducted on CAD cylinder. The basic equation for column buckling is: models and actual lab testing. Sizing of the springs was determined utilizing FEA software (Figure 5  2 EI shows how FEA was utilized to evaluate the stress (2) PCR  2 [5] loading of the springs). One assumption made was (Le ) that the carcass would not provide any addition Where: support and all the support would have to come from Pcr = critical load where buckling occurs the spring stiffness. Upon load deflection testing on E = Young's modulus, 30x10^6psi for most the concept, the load carrying capability was steels significantly greater than computed in the FEA I = moment of inertia. For a round hollow analysis. This meant that the cords in the tire were section: being put in significant tension to support the load of  (OD 4  ID 4 ) the simulated axle loading. Meaning the size and I  (3) Unclassified/Distribution A 64 Approved for Public Release Unclassified/Distribution A Approved for Public Release

Again the stress in the ring can be estimated by replacing the ring as a cylinder and doing the stress equation for a hollow cylinder. The basic equation for critical compressive stress due to these loads:

P Carbon Fiber Hoop Rings   [6] A (4) Steel Belt # 1 Steel Belt # 2 Where:  (OD 2  ID 2 ) Carcass A  (5)

4 Figure 7. Rectangular Hoop Ring Cross-Section

The equations above are going to provide ROM estimates as to the deflection and loading for a specific shape. FEA analysis combined with CAD can provide a much more detailed result and allow for different shapes. The equations do serve as a check to help certify that the boundary conditions in the FEA analysis are correct. The real check is actual test results. These results can be pluged back into the modeling and simulation analysis to provide better theoretical results for design optimization. One such shape considered is the round ring hoop. The one drawback with the shape is that it has little surface area for bonding the carbon fiber to the rubber.

Figure 8. In Process Prototype Carbon Fiber Hoop Ring Molded Tire

Two different military tires were developed using the carbon fiber hoop technology and tested. The most encouraging result was how well the load Figure 6. Round Hoop Ring Cross-Section deflection curve matched the normal pneumatic tire. However, it was able to accomplish this at a much To improve bonding area yet maintain adequate lower air pressure. Fig 9 shows how the load volume between the rings to distribute the loading a deflection of the baseline tire and the tire with carbon more rectangular shape with rounded edges was fiber rings imbedded in the same tire. And Fig 12 developed to maximize the contact area and yet have shows the test results that were achieved on a enough rubber material to distribute the loading. Fig HMMWV tire with carbon fiber hoop technology 7 shows a sketch of what the preferred shape of the imbedded into the tire. The run-flat mileage achieved rings would be for an optimized shape. Fig 8 shows was 800 miles at rated load at a speed of 50 mph. the actual process applied to a current tire. The carbon fiber was added by removing the tread, cutting the grooves for the carbon fiber and then retreading the tire. Unclassified/Distribution A Approved for Public Release Unclassified/Distribution A Approved for Public Release

pneumatic tire. This system enables the tire to retain its working capacity without need to reduce speed or limit travel mileage even when the rubber-cord shell is mechanically damaged due to a puncture, rip or bullet hole. The main advantages of the Hoop Tire over the traditional pneumatic tire and other run-flat solutions are:

 Resistance to physical impacts & mechanical damages  Retention of its physical properties throughout operation  Better thermal characteristics  Durability & safety Figure 9. Load Deflection Curves  No need for pressure sensors or alarm indicators  Explosion of a hoop tire is an impossible For the tire analysis, FEA was used to help baseline event the tire. Fig 10 shows the stress analysis results of  Cost-effective (in compare to run-flat inflating a tire to 50 psi. solutions)

Prototypes & Technology

The hoop tire technology consists of a unique internal system of elastic elements, vulcanized into a rubber-cord shell. Based on a mathematical algorithm developed by a team of scientists models of spring configurations were built and tested in order to define the optimal parameters of the elastic structure, such as: wire diameter, the number of elastic elements & type of materials. Various types Figure 10. FEA of Inflated Tire Model of rubber were tested and special techniques for The important analysis and correlation is assembly & vulcanization were developed. determining the footprint and resulting contact pressure from a given load or tire deflection. An The R&D process was based on FEA analysis was performed assuming a deflection of 1.2 software, using the ‘Finite Elements Method’ (FEM) inches that resulted in a rated load of 4,167 lbs. The to predict following features: weight, radial and resulting FEA analysis (ref fig 3) provided an lateral stiffness, stress level in tire components & average contact pressure of 56 psi, which is very vibration frequencies. close to the inflation pressure of 50 psi. This proves that the FEA model is close to the physical model of 15 Hoop tire prototypes were manufactured the tire and can be used as a start for the more and have undergone a variety of static, dynamic & advanced models that will incorporate the carbon road tests. Based on the tests’ results & findings, a fiber rings. methodology was established for the development of various tires, (types & dimensions), for different R&D applications.

While the R&D of run-flat tires has been based Test Results & Findings on pneumatic tires, (mainly focused on protecting the tires’ side walls or supporting metal rings), with During March, June & December, 2000, 3 limited operational capacity in case of a mechanical sessions of indoor tests were conducted. The tests damage & deflated tire, hoop tire technology is a were conducted on 13 prototypes of hoop tire completely new concept that eliminates the use of models, in comparison to conventional pneumatic pressurized air. It consists of a system of elastic tires. The tests were conducted, even if not to the elements and possesses all the features of a letter, in accordance to the following specifications: Unclassified/Distribution A Approved for Public Release Unclassified/Distribution A Approved for Public Release

shock-absorbing property of a traditional pneumatic  M81 Static Test (for tire stiffness) tire. The circumference of the soft polyurethane foam  H12 DOT Endurance FMV SS 109 Test layer is bonded to a Carbon Fiber ring along with tire  H02 FMVSS 109 High Speed tread. By varying the thickness and geometry of the Performance 5.5.5 polyurethane soft layer, this unique tire-wheel  H24 Plunger Energy Certification – assembly can generate a wide array of ride and GMVSS 109 & 117 handling performance.

The tests results & findings indicate that the MTW’s vertical stiffness (ride comfort hoop tire has approximate, or in some parameters, performance) and lateral stiffness (handling and even better characteristics than the comparative- cornering performance) can both be optimized, tested pneumatic tire. pushing the performance envelope in various military applications. The purpose of this paper is to illustrate The hoop tire technology leads to a puncture the effectiveness of integrated footprint analysis tolerant & lighter weight system, eliminating the using FEA in the designs of “pneumatic tires”, “high need for pressurized air. This unique technology aspect ratio airless tires”, and “solid foam tires”. This might be first targeted at military, off-road & research work is development of a military agricultural vehicles and further developed for light using carbon fiber and Polyurethane. This paper trucks and even passengers’ cars. provides an integral step by step approach to model the footprint analysis using FEA standard software. The continued R&D program of the hoop tire The computer simulation developed in this paper is should be focused and oriented at the following divided into three tire designs according to the type issues: of tire applications.

 Improving the capabilities of the elastic Finite Element Analysis as a Tool for Tire elements by using advanced composite Development materials towards optimal characteristics.  Developing & analyzing various structure Finite Element Analysis (FEA) not only saves time models that are based on the hoop tire’s and resources in the tire development process but elastic elements construction. also provides very good insight into the stress strain distribution of the composite tire. This leads to better  Improving the vulcanization process for a better performance. understanding of the functionalities and failure modes of tire by revealing the critical regions.  Developing prototypes for various sizes & Various studies like stress and deformation of the tire carrying capacity. (Ridha, 1980), transient analysis of tire (Nakajima & Developing a mass-production technology.  Padovan, 1987), rolling analysis (Shiraishi et al.

2000) has been instrumental in the application of The next stage of development is the Military Tire- Finite Element Analysis (FEA) in the Tire design Wheel (MTW) Assembly for future tactical and process. This paper addresses the importance of combat vehicles. This program is placing emphasis footprint analysis in tire design and presents a on handling, traction, and cornering tire performance complete description of tire modeling using for the light trucks. In order to meet these high commercial software. There are several publications performance standards, solid tire with aspect ratio on the tire modeling such as Zamzamzadeh et al, in lower than 0.35 have been developed the literature that provides the outline of the footprint

analysis and its importance. But very few discuss the • Tire Size: LT225/35R19 challenges associated with analysis and the • Rim Width: 7.0 inches precautions needed to overcome non-convergence • Tread Width: 7.6inches issues in the simulation. This paper presents step by • Overall Diameter: 25.0 inches step modeling of the footprint analysis of a • Speed rating:60mph pneumatic tire and further extends the analysis to

“airless” solid tire design. This tire aims to eliminate tire blowouts with its integrated Carbon Fiber Ring-Wheel “MTW” assembly, a solid one-piece wheel-and-tread system that could soon enter manufacturing. The MTW's rim is bonded to soft polyurethane foam that provides the Unclassified/Distribution A Approved for Public Release Unclassified/Distribution A Approved for Public Release

c. Apply the actual static vertical Model Description loading of 2335/2 lb on the half tire d. Reflect the tire to obtain the full 3D This part of the paper presents the design and tire. analysis of a pneumatic radial P265/65 R 17 tire using FEA software. Numerous assumptions were Modeling of 2D half model: made in the model definition to achieve computational advantage by reaching converged Constructing the core of the tire and modeling of solutions in reasonable time. An integrated footprint each component is the first step into the modeling of analysis consists of three steps: the half 2D axisymmetric model. A general tire has the following major components 1) The first step is the modeling of the core of 1. Rubber materials - Tread, Belt Region, Inner tire for the axisymmetric analysis. A Liner, Sidewall Region, Inner Carcass simplified 2D half model can be used as the Region, Bead Filler Region, and base model and later it can be revolved and Apex/Chafer Region. reflected to obtain the full tire model. 2. Reinforced materials - Beads, Nylon Cap Initiating the simulation with a 2D half Ply, Steel Belts, Carcass Ply and Carbon model saves memory and simulation time Fiber Hoop for the analysis. The simplified full tire obtained by this process can be called a The choice of elements and material properties for smooth tire as the tread geometry is kept each component in the tire plays significant role in simple and smooth with a few radial the convergences and simulation time. It is known grooves. The output results from the that a full structural quad mesh is needed for accurate footprint analysis like load-deflection, total results in any FEA analysis. This study includes the footprint area and average contact pressure 2D tire models meshed with lower order tetrahedral of the tire depends on the core and the elements which is capable of producing reliable reinforcements design and does not depend results in less computational time. Applying on the tread design. The actual tread should tetrahedral mesh also saves considerable time that to be used in place of the smooth tread in would be needed to mesh the tire with structural quad order to obtain accurate values of the elements. maximum contact pressure and the total Both the axisymmetric quadrilateral and footprint contact area. axisymmetric triangular elements with twist, are 2) The second step is to mount the 2D tire suitable for meshing. It should be noted that using geometry on a rigid rim and perform full order elements rather than reduced order gives inflation analysis. The standard inflation much faster convergence and accurate results. pressure of 30 psi, 40 psi and 50 psi are used. In this analysis the stress and strain It is a standard practice to use incompressible energy distribution of different tire regions hybrid elements for modeling the rubber regions of can be observed and recorded. the tire. But when undergoing large compressive 3) The third step is to revolve the 2D geometry stress in the area under the belt region elements using the symmetric model generation sometimes force the solver to create convergence (SMG) and obtain half 3D tire model. The problems. Also the hybrid elements have been footprint model can be setup by defining the observed to perform well in 2D model but fails in 3D rigid road using analytical rigid geometry model. Based on the experience gained a and defining the contact interaction of the compressible full order element with Poisson’s ration tire and road. The footprint analysis includes of 0.495 was used. Neo Hooke model and Mooney- 4 load steps: Rivlin model were used for modeling the rubber components. The reinforcements i.e. belts, carcass a. Bring the inflation analysis results plies and nylon chords can be modeled as the surface in equilibrium by transferring the elements with rebar properties defined and embedded results from the 2D analysis to 3D in the belt region. The rebar definition allows the analysis. analysis to control parameters like orientation angles b. Apply displacement controlled of steel belts, diameter and number of strands, loading on the rigid road surface to distance between strands. The bead regions can be establish contact with the tire. modeled as a single cross sectional area of steel embedded inside the rubber to represent the cluster of Unclassified/Distribution A Approved for Public Release Unclassified/Distribution A Approved for Public Release strands of small diameter steel wires. The material property of the bead can not be represented as the material property of the steel as it should represent steel strands embedded in rubber, so the young’s modulus can be modified using a ratio of total cross sectional area of steel wires with the total area of the bead. In this study the Young’s modulus of the bead is assumed to be half of the Young’s modulus of steel. The material property for the Nylon and Textile material for the carcass ply is defined as a Marlow model.

Tire analysis is a pure nonlinear analysis comprising of geometric nonlinearity, material nonlinearity and also nonlinearity arising from contact interaction between tire and rim and tire and road, so the large deformation effects should be included. The rim can be modeled in more than one ways, but in all cases an axisymmetric analytical rigid surface is used. The road is modeled as analytical rigid surface and the interaction between the road and tire is modeled based on a small value of friction (< 0.2). For any type of contact interaction it is a very good practice to use a small initial loading (a) to establish contact in the first step of the analysis (b) and then engage the full load or displacement to Figure 11: (a) Footprint analysis of half tire model complete the analysis. Study has shown that the (b) Contact pressure distribution of the footprint area software needs such a gradual approach to overcome convergence issues during contact analysis. Figure 11 shows the footprint analysis of the half tire model. From this analysis the deflection of this Footprint Analysis of pneumatic tire tire was recorded to be 1.2 inch for the rated load of 4167.5 lbs and inflation pressure of 50 psi. The The footprint analysis consists of inflation and the average contact pressure in the footprint region application of rated load on the tire. The 2D inflation should be same as the internal pressure of the tire, i.e. analysis results can be transferred into a 3D model. inflation pressure. The observed average contact The application of the rated load can be done in two pressure is 56 psi, which is very close to the internal steps, first step is displacement controlled loading, pressure of 50 psi , that proves that this FEA model is where the road can be displaced and brought in close to the physical model of the tire. From test contact with the tire. In the second step the actual results of a similar tires and the footprint analysis load can applied on the road, keeping the rim fixed. If results it is understood that a tire with nominal ride half of the tire model is used, the rated load should be comfort should have the characteristic stiffness half too. During the second step the rim boundary corresponding to displacement of 1.2” to 1.5” for a condition can be kept free in z direction to record the rated load of 4335 lb. Also it is understood that a displacement of the tire center due to the rated load. large value of footprint area is needed for stability of The load-displacement diagram of the tire is one of the tire. the important tools for the verification of the model by using test data. Other parameters important for observation is the total footprint area, average footprint contact pressure and the maximum contact pressure. By recording the stress strain in the compressed footprint region and the tire region just opposite to the footprint region, a fair idea of the periodic values of the stress/strain in the tire can be obtained. These values provide extremely important insight into the performance of the tire.

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A comparison of the footprint analysis of tire is the tire. Conceptual designs of airless tires were provided in Table 1. The three parameters presented modeled and validated very quickly and optimized in this table are footprint area, maximum design selected at the end. Effect of different material displacement and the maximum contact pressure. properties has also been studied and proper combination of material selected for the analysis. The Table 1: Footprint Analysis result shows that run-flat technology can be incorporated into the tire easily that provide the same Model Footprint Max Max ride comfort as normal pneumatic tires. Rated load: Area Displacement Contact 4335 lb (sq inch) (inch) Pressure A discussion of finite element analysis applied to (psi) tires was included to show how various design Pneumatic tire 41.48 1.2 90.37 parameters of a non-pneumatic tire can be optimized in new designs rapidly. The run-flat technology is Conclusion built directly into the tire, yet maintains the normal variable footprint of a pneumatic tire. This makes the This paper discusses a number of unique ideas for tire/wheel assembly lighter and far more survivable the development of run flat tires to enhance the than normal military run-flat technology. Logistical, survivability of military vehicles in a variety of economic, and fuel economy are additional benefits terrains and theatres throughout the world. The this tire technology provides over current military ability to travel in as much varying terrain as possible tires. This new tire technology has been developed provides the military greater tactical options. This that substantially increases survivability, eliminates requirement/need is for the tire to provide a variable the need for heavy run-flat inserts, significantly footprint to allow for different ground pressure. The reduces air pressure requirements and provides full displacement of the tire with rated load and the (or near full) speed capability in degraded/damaged footprint area in contact with the road surface was mode (punctured tire). found to be a very good parameter to be used as the design criterion. The deflection of the tire corresponds to the stiffness of the tire and the footprint area corresponds to the ride comfort of a performance tire. Commercial software has been used as the modeling tool to prove the effectiveness of the integrated footprint analysis for the design process of

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Figure 12. In Process Prototype Carbon Fiber Hoop

Ring Molded Tire

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REFERENCES [1] Runflat International Limited, “Runflat-Runflat Tyre Systems.” http://www.army-technology.com/contractors/tracks/runflat/ Accessed 15 July 2010. [2] HEADQUARTERS, DEPARTMENTS OF THE ARMY,THE AIR FORCE, AND MARINE CORPS, October 1997, ARMY TM 9-2320-387-10. [3] Coustan, Dave. "How Strykers Work." 17 September 2004. HowStuffWorks.com. http://science.howstuffworks.com/stryker.htm 15 July 2010. [4] Bilek, Mark. "7 Tire Buying Tips." 21 October 2005. HowStuffWorks.com. http://auto.howstuffworks.com/buying-selling/cg-tire-buying-tips.htm 15 July 2010. [5] Young, Warren, Richard Budynas, Roark’s Formulas for Stress and Strain, Seventh Edition. New York: McGraw Hill, 2002. [6] Beer, Ferdinand, E. Russell Johnston, Jr., Mechanics of Materials. New York: McGraw Hill, 1981. [7] Nakajima, Y., and Padovan, J. 1987. Finite element analysis of steady and transiently moving/rolling nonlinear viscoelastic structure--Iii. Impact/contact simulations. Computers and Structures 27(2):275-286. [8] Ridha, R.A. 1980. Computation of stresses, strains, and deformations of tires. Rubber Chemistry and Technology 53(4):849-902 [9] Shiraishi, M.,Yoshinaga, H., Miyori, A., and Takahashi, E. 2000. Simulation of Dynamically Rolling Tire. Tire Science and Technology 28(4):264-276. [10] Zamzamzadeh, M., and Negarestani, M. 2006. A 3D Tire/Road Interaction Simulation by a developed Finite Element Model. Tire Society Conference, 11-12 Sep. 2006, Akron, USA.

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